KS3 · Maths

Finding the equation of a straight line (y = mx + c)

y = 2x + 5 hides two numbers: one says how steep the line is, the other says where it crosses the y-axis. Spot them and you know the line.

Slide the line y = 2x + c

01234-8081624xy1y when x = 0
the constant c 1

the constant c: 1. y when x = 0: 1

Every line here is y = 2x + c, so for every 1 you move right, the line goes up 2. Drag the dot on the y-axis to change c. Try c = 5, c = 1 and c = −1. Where does each line cross the y-axis — and does the line ever get steeper?

Algebra · y = mx + c

Ready to read, or rearrange first?

Can you read the gradient and the y-intercept straight off this equation?

Still to sort

Read m and c straight off (0)

y is on its own, with a coefficient of 1. The other side is just an x-term and a number, in either order.

Where the line is: A missing number still counts: no number in front of x means 1x, and no constant means the constant is 0.

Rearrange first (0)

y has a coefficient that isn't 1, there's a bracket, or y isn't on its own.

Where the line is: These are still straight lines. They just aren't showing you m and c yet.

8 of 8 still to sort.

You can only read m and c straight off when the equation says y = (number)x + (number), with y on its own. Some equations are already there. Some are in disguise.

Maths · Algebra

Get y on its own, then read it

Do the same thing to both sides until the equation reads y = mx + c. Pick an equation and step through it.

GoalFind the gradient and y-intercept of 2y = 10 − 4x
1
2y = 10 − 4x
Start

y has a coefficient of 2, so these numbers aren't m and c yet.

2
3

Step 1 of 3

y has a coefficient of 2, so these numbers aren't m and c yet.

Watch out: Divide EVERY term. If you halve 2y = 10 − 4x but forget the 10, you get y = 10 − 2x — a different line, crossing the y-axis at (0, 10) instead of (0, 5).

From a graph

Spot the slip

A straight line is drawn on a grid. On the x-axis each square is worth 2; on the y-axis each square is worth 1. The line crosses the y-axis at (0, 3). Following the line from there, you land exactly on a grid corner after 5 squares across and 4 squares up. Write the equation of the line.

A student's answer — which line goes wrong?

Predict, then check

Look closely: x = 0 isn't in this table.

These values come from a straight line. x: 2, 3, 4, 5 → y: 4, 8, 12, 16. Which equation is it?

From two points

Your turn: fill in the missing steps

Find the equation of the line through (3, 4) and (5, 8).

  1. Quick sketch: going right from (3, 4) to (5, 8), the line goes up — so expect a positive gradient.
  2. Change in y = 8 − 4 = 4. Change in x = 5 − 3 = 2.
  3. missing step
Which line is step 3?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

y = mx + c

Two numbers pin a straight line down: how steep it is, and where it crosses the y-axis.

What you need to know

  • In y = mx + c, the coefficient of x, m, is the gradient: how steep the line is.
  • The constant c gives the y-intercept (0, c), because on the y-axis x = 0, so mx = 0 and y = c.
  • You can only read m and c straight off once y is on its own with a coefficient of 1 — otherwise rearrange first.
  • Once lines are in the form y = mx + c, the same coefficient of x means parallel lines; different constants just move them up or down.
  • To write a line's equation, find m and c (from a graph, a table or two points), then check by substituting.

The big picture

When a line's equation is written as y = mx + c, the coefficient of x (m) is the gradient and the constant (c) gives the y-intercept (0, c). That only works once y is on its own, so some equations need rearranging first. Working the other way, you can write a line's equation by finding m and c — from a graph, a table of values or two points — and check it by substituting.

Key points

1Hidden values count: y = x − 2 has gradient 1, y = 2x has y-intercept (0, 0), and y = 5 is a horizontal line with gradient 0.
2Terms can come in either order: y = 5 − 2x has gradient −2 and y-intercept (0, 5).
3Rearrange by doing the same to both sides: 2y = 10 − 4x becomes y = 5 − 2x; y = 2(3x − 4) becomes y = 6x − 8.
4Knowing m and c is enough to write the line: gradient 4 and y-intercept (0, 7) give y = 4x + 7.
5From a graph, read the changes off the axis numbers, not by counting squares. If the line goes down as you move right, the gradient is negative: down 2 for every 1 right, crossing at (0, 3), gives y = −2x + 3.
6From a table or two points: gradient = change in y ÷ change in x, then walk back to x = 0 to find c.

Worked example

Problem

Line A is 4x + 2y = 6. Line B passes through (1, 3) and (3, −1). Are the two lines parallel? Are they the same line?

⚠ Watch out

Reading the numbers off before y is on its own. In 2y = 10 − 4x the gradient is not −4 and the y-intercept is not (0, 10). Divide every term by 2 first: y = 5 − 2x, so the gradient is −2 and the y-intercept is (0, 5).

🧠

Memory hook

Start at c, climb m. The line starts on the y-axis at (0, c); then for every 1 step right, it climbs m (a negative m means it goes down).

✓

Check yourself

Cover the page and try 3y = 6x + 12: rearrange it, name its gradient and y-intercept, and write a line parallel to it. Check the intercept by putting x = 0 in.

Flashcards

(16)
In y = mx + c, what does m tell you?
The gradient — how steep the line is. m is the coefficient of x.
In y = mx + c, what does c tell you?
Where the line crosses the y-axis: the y-intercept is (0, c).
Why is the constant c the y-intercept?
On the y-axis x = 0, so mx = 0 and y = c.
What is a coefficient? What is a constant?
A coefficient is the number multiplying a variable (3 in 3x). A constant is a term with no variable (−5 in y = 3x − 5).
When can you read m and c straight off an equation?
Only when it's in the form y = mx + c: y on its own, with a coefficient of 1.
Gradient and y-intercept of y = 5 − 2x?
Gradient −2, y-intercept (0, 5). The terms are just the other way round.
Gradient of y = x − 2?
1 — no number in front of x means one x. (Its y-intercept is (0, −2).)
y-intercept of y = 2x?
(0, 0). No constant means the constant is 0, so the line goes through the origin.
What kind of line is y = 5?
A horizontal line: gradient 0, y-intercept (0, 5).
Gradient and y-intercept of y = 2(3x − 4)?
Expand first: y = 6x − 8. Gradient 6, y-intercept (0, −8).
y = 2x + 5 and y = 2x − 1: what do they share, and what's different?
Same gradient (2), so they're parallel. Different y-intercepts: (0, 5) and (0, −1).
Gradient 4, y-intercept (0, 7). Equation?
y = 4x + 7
How do you find the gradient from two points?
Change in y ÷ change in x.
A table of values doesn't include x = 0. How do you find c?
Work backwards through the gradient, one step at a time, until x = 0.
Reading a gradient from a graph: what must you check first?
The scales on both axes. A change of 10 in x and 4 in y gives 4 ÷ 10 = 2/5, however many squares that is.
How do you check an equation you've found?
Substitute each point's x and see if you get its y. Every point on the line must fit.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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